Fxn

Fxn is a shorthand commonly used to denote a function, particularly in mathematical notes and programming contexts. While the symbol f is the standard name for a function, fxn may appear as a variable identifier or informal label for a function in code or teaching materials.

Formally, a function f from a set X (the domain) to a set Y (the codomain) assigns to every element x in X exactly one element f(x) in Y. The pair (x, f(x)) is a member of the graph of f, and the image of X under f is the subset {f(x) : x ∈ X} of Y. The domain and codomain can be the same set or different ones.

Key properties of functions include injectivity, surjectivity, and bijectivity. A function is injective (one-to-one) if different inputs map to different outputs. It is surjective (onto) if every element of the codomain is an output for some input. A function that is both injective and surjective is bijective, and it has an inverse function defined on the codomain. Function composition combines functions to form new ones, and the identity function id_X maps each element to itself.

Examples help illustrate the ideas. Consider f(x) = x^2 with domain R and codomain R; this f is not injective because f(a) = f(-a). If the domain is restricted to x ≥ 0, f becomes bijective onto [0, ∞). Another example is f: N → N with f(n) = 2n, which is injective but not surjective.

In programming, fxn or function refers to a named block of code that accepts inputs (parameters) and returns an output. Functions can be first-class values, passed as arguments, stored in variables, or returned from other functions in many programming languages.